If momentum is increased by 20% , then K.E. increase by

Options

A. 0.55
B. 0.77
C. 0.66
D. 0.44

Explanation

To solve the problem of how much the kinetic energy (K.E.) increases when momentum is increased by 20%, we will follow these steps:

### Step 1: Understand the relationship between momentum and kinetic energy
The formula for kinetic energy (K.E.) in terms of momentum (P) is given by:
\[ K.E. = \frac{P^2}{2M} \]
where \(M\) is the mass of the object.

### Step 2: Define the initial momentum
Let the initial momentum be \(P\). Therefore, the initial kinetic energy \(K\) can be expressed as:
\[ K = \frac{P^2}{2M} \]

### Step 3: Calculate the new momentum after a 20% increase
If the momentum is increased by 20%, the new momentum \(P'\) can be calculated as:
\[ P' = P + 0.2P = 1.2P \]

### Step 4: Calculate the new kinetic energy with the new momentum
Now, we can find the new kinetic energy \(K'\) using the new momentum:
\[ K' = \frac{(P')^2}{2M} = \frac{(1.2P)^2}{2M} \]
Calculating this gives:
\[ K' = \frac{1.44P^2}{2M} \]

### Step 5: Find the change in kinetic energy
The change in kinetic energy \(\Delta K\) is given by:
\[ \Delta K = K' - K \]
Substituting the expressions for \(K'\) and \(K\):
\[ \Delta K = \frac{1.44P^2}{2M} - \frac{P^2}{2M} \]
Factoring out \(\frac{P^2}{2M}\) gives:
\[ \Delta K = \frac{P^2}{2M}(1.44 - 1) = \frac{P^2}{2M} \cdot 0.44 \]

### Step 6: Calculate the percentage increase in kinetic energy
To find the percentage increase in kinetic energy, we use the formula:
\[ \text{Percentage Increase} = \left(\frac{\Delta K}{K}\right) \times 100\% \]
Substituting the values:
\[ \text{Percentage Increase} = \left(\frac{0.44 \cdot \frac{P^2}{2M}}{\frac{P^2}{2M}}\right) \times 100\% = 0.44 \times 100\% = 44\% \]

### Final Answer
Thus, when momentum is increased by 20%, the kinetic energy increases by **44%**.

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